In other words, each of the 16 SM chemical potentials, \(\mu _i\)  (\(i = e,\mu \,\tau ,\ell _e,\ell _\mu ,\ell _\tau ,u,c,t,d,s,b,Q_1,Q_2,Q_3,\Phi \) ), can be expressed in terms of a linear combination of the nine chemical potentials \(\bar{\mu }_C\)  (\(C = u,B,\tau ,\mu ,u-d,e,\Delta _e,\Delta _\mu ,\Delta _\tau \) ).\(11\endcsname \)Here, the bar over the chemical potentials associated with global charges, \(\bar{\mu }_C\) , indicates that all internal gauge degrees of freedom have been summed over, which is not the case for the ordinary chemical potentials \(\mu _i\) .
As a consequence, we have, e.g., \(\mu _u = {1}{3}\,\bar{\mu }_u\) .
This distinction is not necessary for quantities like the charge densities \(q_i = g_i \mu _i T^2/6\)  and \(q_C = \bar{\mu }_C T^2/6\)  or the charge-to-photon ratios \(\eta _i = q_i/n_\gamma \)  and \(\eta _C = q_C/n_\gamma \) , all of which include the same \(g_i\)  factors.
For more details on our conventions, see Ref. [1]}.

In order to work out these linear combinations, we employ the linear-algebra formalism developed in the appendix of Ref. [1]}.
The essence of this formalism is to write down a system of 16 linear equations — one constraint equation for each conserved charge and one equilibrium condition for each interaction in chemical equilibrium — which can be solved for the 16 SM chemical potentials.
Working with 15 constraint equations and one equilibrium condition for the top-quark Yukawa interaction, we thus obtain
\(\begin{pmatrix}\mu _e \\ \mu _\mu \\ \mu _\tau \\ \mu _{\ell _e} \\ \mu _{\ell _\mu } \\ \mu _{\ell _\tau } \\ \mu _u \\ \mu _c \\ \mu _t \\ \mu _d \\ \mu _s \\ \mu _b \\ \mu _{Q_1} \\ \mu _{Q_2} \\ \mu _{Q_3} \\ \mu _\Phi \end{pmatrix}=\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & {1}{6} & 0 & 0 & 0 & -{1}{2} & -{1}{2} & 0 & 0 \\0 & {1}{6} & 0 & -{1}{2} & 0 & 0 & 0 & -{1}{2} & 0 \\0 & {1}{6} & -{1}{2} & 0 & 0 & 0 & 0 & 0 & -{1}{2} \\{1}{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\{1}{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\{1}{27} & {2}{27} & {1}{9} & {1}{9} & -{7}{27} & {1}{9} & -{1}{9} & -{1}{9} & -{1}{9} \\{1}{3} & 0 & 0 & 0 & -{1}{3} & 0 & 0 & 0 & 0 \\{1}{3} & 0 & 0 & 0 & -{1}{3} & 0 & 0 & 0 & 0 \\{1}{3} & 0 & 0 & 0 & -{1}{3} & 0 & 0 & 0 & 0 \\-{1}{3} & {1}{6} & 0 & 0 & {1}{6} & 0 & 0 & 0 & 0 \\-{1}{3} & {1}{6} & 0 & 0 & {1}{6} & 0 & 0 & 0 & 0 \\-{5}{27} & {7}{54} & -{1}{18} & -{1}{18} & {8}{27} & -{1}{18} & {1}{18} & {1}{18} & {1}{18} \\{2}{9} & -{1}{18} & {1}{6} & {1}{6} & -{5}{9} & {1}{6} & -{1}{6} & -{1}{6} & -{1}{6}\end{pmatrix}\begin{pmatrix}\bar{\mu }_u \\ \bar{\mu }_B \\ \bar{\mu }_\tau \\ \bar{\mu }_\mu \\ \bar{\mu }_{u-d} \\ \bar{\mu }_e \\ \bar{\mu }_{\Delta _e} \\ \bar{\mu }_{\Delta _\mu } \\\bar{\mu }_{\Delta _\tau }\end{pmatrix} \,,\) 
